The n-solvable filtration, defined by Cochran, Orr, and Teichner in the late 90's, gives structure to the smooth knot and link concordance groups. Much is known about the n-solvable filtration of the knot concordance group for small n. For example, a knot is 0-solvable if and only if it has Arf invariant zero. Moreover, a knot is 0.5-solvable precisely when it's Seifert matrix looks like that of a slice knot, called algebraically slice. However, very little is known for links. In this talk, we will completely classify 0-solvable links and discuss recent progress towards understanding 0.5-solvable links.